Ok and our understandings are different.
I'm aware, in fact my point was that yours, and the American understanding more generally, are problems.
It being problematic in a case like this is an opinion.
You can call what anyone thinks about society and what should be done in it an opinion. That doesn't have very much analytical value though.
Provides closure, stops the person from ever possibly being a threat to anyone again (people in jail are still threats to workers and other inmates).
But to do so someone must be killed, something we clearly agree is a bad thing. That's a heavy cost isn't it? I mean that's the reason your suggesting someone be killed in this case.
What's interesting then, and the root of the issue, is why you think killing someone leads to the state needing to kill them.
This is of course ignoring the extremely real, and clearly racial and classist in implantation, chance that the state will get this wrong.
Rehabilitation and incapacitation are only one of the components of imprisonment in the justice system. There are also retribution and deterrence - the two that are typically wilfully ignored by some people.
Are they willfully ignored? I don't think anyone ignores or has an issue with deterrence. Though vindictive people seem to vastly overestimate its effect as a justification for what they are really after.
Now the latter people aren't willfully ignoring, the other position is that retribution should not be part of the justice system. It's a relic of worse times.